非自治/随机系统的渐近性态及其应用

项目名称： 非自治/随机系统的渐近性态及其应用

项目编号： No.11201226

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 曹峰

作者单位： 南京航空航天大学

项目金额： 22万元

中文摘要： 本项目旨在综合运用微分方程与动力系统的多个分支，包括斜积半流理论，无限维光滑动力系统，随机动力系统，Lyapunov指数以及遍历理论等，来研究（强）单调非自治/随机系统轨道的渐近对称性以及无界域上几乎周期抛物方程的"一致衰减"解的渐近收敛性。在condensing映射的范畴下，研究抽象强单调非自治系统稳定解关于紧连通群的渐近对称性，并应用于带扩散项的有限和无限时滞泛函微分方程以及部分扩散系数为零的反应扩散系统得到其稳定解的渐近对称性；进而通过对无界域上非自治几乎周期抛物方程"一致衰减"解的极限集的研究，一般性地建立系统解的渐近几乎自守或者几乎周期性；最后利用Lyapunov指数理论和指数分离性质，研究由random抛物方程生成的（强）单调随机系统的稳定解关于紧连通群的渐近对称性。

中文关键词： 非自治系统；抛物方程；收敛性；渐近对称性；

英文摘要： We intend to investigate the asymptotic symmetry in (strongly) monotone non-autonomous/random systems and the convergence for uniform decay solutions of almost periodic parabolic equations on unbounded domains. Many branches of differential equations and dynamical systems will be involved, such as skew-product semiflows theory, infinite-dimensional dynamical systems, random systems, Lyapunov exponents and ergodic theory. Firstly, in the context of condensing maps, we will study the asymptotic symmetry, with respect to some compact connected group, for stable solutions in strongly monotone non-autonomous systems. We then apply this theory to functional differential equations with finite/infinite delay and spatial diffusion, and reaction-diffusion systems with partly vanished diffusion coefficients to study their asymptotic symmetry. Secondly, we will investigate the limit sets of the uniform decay solutions of almost periodic parabolic equations on unbounded domains, and establish their asymptotic almost automorphy or almost periodicity. Finally, we will utilize the theory of exponential separation and Lyapunov exponents to study the asymptotic symmetry in (strongly) monotone random systems generated by random parabolic equations.

英文关键词： Non-autonomous systems；Parabolic equations；Convergence；Asymptotic symmetry；